U.S. patent number 5,260,882 [Application Number 07/636,629] was granted by the patent office on 1993-11-09 for process for the estimation of physical and chemical properties of a proposed polymeric or copolymeric substance or material.
This patent grant is currently assigned to Rohm and Haas Company. Invention is credited to Mario Blanco, Thomas H. Pierce.
United States Patent |
5,260,882 |
Blanco , et al. |
November 9, 1993 |
Process for the estimation of physical and chemical properties of a
proposed polymeric or copolymeric substance or material
Abstract
A process for estimation of properties utilizing experimental
information using constraint determined by chemical kinetics,
statistical thermodynamics and molecular mechanics including
experimental information on proposed polymeric or copolymeric
substances of large molecules for the estimation of the physical
properties of the substances by first defining the substances
molecular chemical composition, second, estimating properties of
the molecular chemical composition when 3-dimensionally folded,
third, forming the composition into a polymeric cluster, fourth,
estimating the physical properties of the polymeric cluster, and
finally preparation of the polymeric substances having the
properties as estimated. The present invention overcomes the
"multiple stable minimum " which is associated with prior polymer
modeling approaches of large molecules.
Inventors: |
Blanco; Mario (Jeffersonville,
PA), Pierce; Thomas H. (Lawrenceville, NJ) |
Assignee: |
Rohm and Haas Company
(Philadelphia, PA)
|
Family
ID: |
24552695 |
Appl.
No.: |
07/636,629 |
Filed: |
January 2, 1991 |
Current U.S.
Class: |
703/6;
703/12 |
Current CPC
Class: |
G16C
60/00 (20190201); G16C 20/30 (20190201); C07K
1/00 (20130101) |
Current International
Class: |
C07K
1/00 (20060101); G06F 17/50 (20060101); G06F
015/60 (); G06F 015/20 () |
Field of
Search: |
;364/496,497,499,578,518-522 ;436/86,89 ;935/87
;395/100,119,920,932 |
References Cited
[Referenced By]
U.S. Patent Documents
Other References
Puleo, A. C., Muruganandam, N., Paul, D. R., "Gas Sorption and
Transport in Substituted Polystyrenes", J. of Polymer Science Part
B: Polymer Physics, 1989, 27, 2385-2406 (1989). .
Molecular Silverware. I. General Solutions to Excluded Volume
Constrained Problems, Mario Blanco, Journal of Computational
Chemistry, accepted for publication Jul. 3, 1990. .
MacroModel-An Integrated Software System for Modeling Organic and
Biorganic Molecules Using Molecular Mechanics, by Fariborz Mohamadi
et al, Department of Chemistry, Havemeyer Hall, Columbia
University, New York, N.Y. 10027. .
Statistical Mechanics of Chain Molecules, Paul J. Flory, J. G.
Jackson-C. J. Wood Professor of Chemistry, Stanford University,
Interscient Publishers. .
Alfrey, T. Jr., & C. C. Price; J of Polymer Science, 2, 101
(1947). .
Blanco, M., J. of Computational Chemistry, "The Modeling of
Non-Crystalline Condensed Phases-General Solutions to Excluded
Volume Constrained Problems"-Jul. 3, 1990. .
Clarke, J. "New Opportunities for Modelling Polymers", Chemistry
and Industry, 780-786, Dec. 3, 1990. .
Flory, P. J., "Spatial Configuration of macromolecular Chains",
Nobel Lecture, Dec. 11, 1974, Selected Works of Paul J. Flory,
Mandelken, Mark, Suter, Yoon Eds. Stanford Univ. Press, V. 1. (5),
1985. .
Jean Guillot (Makromol. Chem., Macromol. Symp., 35/36, 269-289
(1990). .
Skeist, I.: J. of the American Chem. Soc., 68, 1781 (1946). .
Sorensen, R. A., Liau, W. B., Boyd, R. H., "Prediction of Polymer
Structures and Properties":, Macromolecules, 21, 194-199, 1988.
.
Walling and Briggs; J. of the American Chem. Society 67, 17741
(1945)..
|
Primary Examiner: Teska; Kevin J.
Attorney, Agent or Firm: Connolly & Hutz
Claims
What is claimed is:
1. A process relating to polymer production wherein, with the aid
of a digital computer, stable polymeric substances are simulated by
empirical functions, an appropriate polymer having useful
properties of size, shape, composition, volume, stiffness, and
polymer structure is selected from said simulated polymeric
substances, and a useful polymer is produced, comprising
providing the computer with a data base of polymer molecular
connectivities of polymeric substances,
from the molecular connecivities selecting information to form a
monomer and determining chemical composition through estimation of
individual polymer chain chemical composition,
calculating by statistical thermodynamics the shape of
three-dimensional folding large molecules of the polymeric
substances,
assembling the resulting 3-dimensionally folded molecular chemical
composition into a proposed polymeric cluster of the polymeric
substances,
visually displaying by computer generated graphics or pictures said
polymeric clusters,
estimating at least one of the physical properties of size, shape,
composition, volume, stiffness or polymer structure of the
resulting proposed displayed polymeric cluster,
choosing a proposed polymeric cluster having at least one useful
physical property,
producing said selected cluster by synthesizing the resulting
polymeric product by a format,
wherein the estimated physical property of the simulated cluster of
the polymeric substance is related to the resulting polymer
product.
2. The process of claim 1, wherein a physical property, i.e. the
effect of a solvent, on the molecular chain chemical composition
and 3-dimensional folding is estimated through In Situ
optimization.
3. The process of claim 1, wherein said assembling step further
comprises use of molecular excluded volume constraints determined
by vector geometry.
4. The process of claim 1, wherein said assembling step further
comprises iterative use of numerical methods.
5. The process of claim 1, wherein said polymer product is a
plastic, packaging material, optical disc material, barrier
membrane, adhesive, viscosity improver, dispersant, electronic
chemical, coating, or synthetic biopolymer.
6. The process of claim 1, wherein said polymer product is a
polymer blend compatibilizer, high temperature plastic,
thermoplastic polymer, thermoplastic polymer blend, elastomeric
polymer, elastomeric polymer blend, amorphous polymer, amorphous
polymer blend, crystalline polymer, crystalline polymer blend,
liquid crystalline polymer or liquid crystalline polymer blend.
7. The process of claim 1, wherein said polymer product is a
barrier membrane formed from bio-separator material.
8. A process relating to polymer production as claimed in claim 1
wherein said polymeric substance is a copolymeric substance.
9. A process relating to polymer production as claimed in claim 1
wherein in the calculation by the method of statistical
thermodynamics the polymeric substance has greater than five (5)
monomer units.
10. A process relating to polymer production as claimed in claim
1,
wherein determining chemical composition through a estimation of
individual polymer chain composition comprises
a function of conversion and reaction time using chemical kinetics
rate expressions.
11. A process relating to polymer properties as claimed in claim
1
wherein said calculating by the method of statistical
thermodynamics comprises a numerical method.
12. In the process as claimed in claim 1 the step of developing a
format for synthesizing the resulting and related polymer
product.
13. A process relating to polymer production by identification of
polymeric substances by empirical functions to simulate final
stable polymeric substances,
with the aid of a digital computer,
comprising providing the computer with a data base of polymer
molecular connectivities of polymeric substances,
from the molecular connectivities selecting information to form a
monomer and determining chemical composition through estimation of
individual polymer chain chemical compositions,
calculating by a method of statistical thermodynamics the shape of
three-dimensional folding large molecules of the polymeric
substances,
assembling the resulting 3-dimensionally folded molecular chemical
composition into a proposed polymeric cluster of the polymeric
substances,
visually displaying, using computer graphics, the computer
generated picture of the resulting polymeric cluster,
evaluating properties of the resulting displayed polymeric
cluster,
using said properties of the visually displayed polymeric cluster
to select an appropriate polymer product,
and synthesizing said selected appropriate polymer product by a
format for synthesizing said polymer product,
wherein the produced polymer product has at least one related
property to the graphically displayed and selected proposed
polymeric structure, whereby final stable polymeric substances are
simulated and produced.
14. A process relating to polymeric production as claimed in claim
13, wherein said polymeric substance is a copolymeric
substance.
15. A process relating to polymer production as claimed in claim
13, wherein in the calculation by the method of statistical
thermodynamics the polymeric substance has greater than five (5)
monomer units.
16. A process relating to polymer production as claimed in claim
13,
wherein determining chemical composition through an estimation of
individual polymer chain composition comprises
a function of conversion and reaction time using chemical kinetics
rate expressions.
17. A process relating to polymer properties as claimed in claim
13
wherein said calculating by the method of statistical
thermodynamics comprises a numerical method.
Description
BACKGROUND FOR THE INVENTION
1. Field Of The Invention
The present invention relates to a process for the estimation and
development of proposed chemical polymeric or copolymeric materials
or substances. In particular, this invention utilizes chemical
kinetics, statistical thermodynamics and molecular mechanics,
including experimental information, in the creation of a more
exacting molecular model.
2. Description Of The Related Art
In the art of modeling polymers, a set of current techniques
involve the use of molecular mechanics or molecular dynamics. Both
methods have proven to be very useful in accounting for the
conformational energetics and properties of polymer molecules.
These methods are based on the premise that a molecule can be
simulated by empirical transferable energy functions that represent
bond stretching, bending, and twisting as well as more distant
non-bonded or steric interactions. Electrostatic forces are
included when appropriate. A stable conformation is found as a
minima in the total energy function. This process has reached a
high state of development and has had many successes. See Sorensen,
R. A., Liau, W. B., Boyd, R. H., "Prediction of Polymer Structures
and Properties", Macromolecules, 21, 194-199, 1988. However, in
contrast to the handful of stable minima found in small molecules,
which are usually indicative of compounds used in the
pharmaceutical or agricultural chemical industries, large polymer
molecules have a great number of such "stable minima".
Consequently, prior art modeling techniques were unable to
effectively model large and complex polymers.
Moreover, except for a few crystalline homopolymers, past polymer
modeling approaches have always divorced the specific chemical
molecular structure, such as atom connectivity and atomic
coordinates of the polymer, from the chemical calculations. It is
both important and desirable to include information about the
chemical molecular structure because a large class of commercial
polymers contain more than one monomer type. These polymers are
often prepared with great regard for the intrinsic differences in
the reaction kinetics of the various monomers.
It is an object of the present invention to provide a process for
modeling polymers which would be effective for large molecules
overcoming the "multiple stable minima" problem, as well as
combining the specific chemical molecular structure in the
calculation of physical and chemical properties. These large
molecules are polymeric or copolymeric substances or materials
which can be used in plastics, packaging materials, optical disk
materials, barrier membranes, adhesives, viscosity improvers,
dispersants, electronic chemicals, coatings, or synthetic
biopolymers. Examples of plastics include polymer blend
compatibilizers, high-temperature plastics, thermoplastic,
elastomeric, amorphous, crystalline or liquid crystalline polymers,
polymer blends, and barrier membranes for bio-separation materials.
FIG. 1 is a computer generated three-dimensionally folded full
molecular structure using the process of the instant invention. All
atoms are included for an atactic methyl methacrylate (MMA) chain,
the main constituent in PLEXIGLAS.RTM., a polymer sold by Rohm
& Haas Company.
Additional objects and advantages of the invention will be set
forth in the description which follows, and in part will be obvious
from the description, or may be learned by practice of the
invention. The objects and advantages of the invention may be
realized and obtained by means of the instrumentalities and
combinations particularly pointed out in the appended claims.
SUMMARY OF THE INVENTION
To achieve the foregoing objects, and in accordance with the
purpose of the invention as embodied and broadly described herein,
there is provided a process for the estimation of
physical properties of a proposed polymeric or copolymeric
substance or material comprising the steps of:
(a) defining or determining the proposed polymeric or copolymeric
substance or material's molecular chemical composition through
estimation of individual polymer chain chemical composition;
(b) estimating properties of the molecular chemical composition
when 3-dimensionally folded;
(c) assembling the resulting 3-dimensionally folded molecular
chemical composition into a polymeric of copolymeric cluster;
and
d) estimating the physical properties of the resulting polymeric of
copolymeric cluster.
In various preferred embodiments, step (a) involves estimating the
molecular chemical composition as a function of conversion using
kinetics rate theory; step (b) uses numerical methods based upon
statistical thermodynamics such as Global Growth, In Situ Growth,
Simple Phantom and Conditional Phantom Growth; step (c) uses
molecular excluded volume constraints determined by vector
geometry, and iterative use of numerical methods such as Global
Growth, In Situ Growth, Simple Phantom, and Conditional Phantom
Growth; and step (d) involves calculation of molecular energy
expressions, or quantitative structure-property relationships such
as thermal properties, optical properties, diffusion properties,
and mechanical properties or a combination thereof.
The instant process utilizes experimental information from chemical
kinetics (reactivity ratios) and statistical thermodynamics
(Boltzmann probabilities) to add physical as well as chemical
constraints in the modeling of the polymer. As a result of the use
of this invention, the modeling of polymers will rely less heavily
upon the intuition and expertise of the polymer modeler and more on
the experimental information at hand.
BRIEF DESCRIPTION OF THE DRAWINGS
The accompanying drawings, which are incorporated in and constitute
a part of the specification, illustrate aspects of the invention
and, together with the general description given above and the
detailed description of the invention given below, serve to explain
the principles of the invention.
FIG. 1. is a computer generated picture of a three-dimensionally
folded model of atactic polymethylmethacrylate;
FIG. 2. is a computer generated picture of a three-dimensionally
folded model of syndiotactic polymethylmethacrylate;
FIG. 3. a computer generated picture of a three-dimensionally
folded model of isotactic polymethylmethacrylate;
FIG. 4. is a schematic diagram illustrating the method for defining
or determining the proposed polymeric or copolymeric substance or
material's molecular chemical composition;
FIG. 5. is a schematic diagram illustrating the Global Growth
method;
FIG. 6. is a schematic diagram illustrating the In Situ Growth
method.
FIG. 7. is a schematic diagram illustrating the Simple Phantom
Growth method;
FIG. 8. is a schematic diagram illustrating the Conditional Phantom
Growth method;
FIG. 9. is a schematic diagram illustrating the polymer assembly
step using vector geometry;
FIG. 10. is a schematic diagram illustrating the multi-chain growth
method for simulating glassy polymer growth;
FIG. 11. is a computer generated picture of the resulting polymer
wherein the starting materials were a 50:50 mixture of methyl
acrylate (MA) and styrene (STY) in accordance with Example 1, case
a;
FIG. 12. is a computer generated picture of the resulting polymer
wherein the starting materials were a 50:50 mixture of methyl
acrylate (MA) and styrene (STY) at 0% conversion in accordance with
Example 1, case a;
FIG. 13. is a computer generated picture of the resulting polymer
wherein the starting materials were a 50:50 mixture of methyl
acrylate (MA) and styrene (STY) at 20% conversion in accordance
with Example 1, case a;
FIG. 14. is a computer generated picture of the resulting polymer
wherein the starting materials were a 50:50 mixture of methyl
acrylate (MA) and styrene (STY) at 40% conversion in accordance
with Example 1, case a;
FIG. 15. is a computer generated picture of the resulting polymer
wherein the starting materials were a 50:50 mixture of methyl
acrylate (MA) and styrene (STY) at 60% conversion in accordance
with Example 1, case a;
FIG. 16. is a computer generated picture of the resulting polymer
wherein the starting materials were a 50:50 mixture of methyl
acrylate (MA) and styrene (STY) at 80% conversion in accordance
with Example 1, case a;
FIG. 17. is a computer generated picture of the resulting polymer
wherein the starting materials were a 50:50 mixture of methyl
acrylate (MA) and styrene (STY) at 99% conversion in accordance
with Example 1, case a;
FIG. 18. is a computer generated picture of the resulting polymer
wherein the starting materials are a 23.3:76.7 mixture of methyl
acrylate (MA) and styrene (STY) in accordance with Example 1, case
b;
FIG. 19. is a computer generated picture of the resulting polymer
wherein the starting materials are a 23.3:76.7 mixture of methyl
acrylate (MA) and styrene (STY) at 99% conversion in accordance
with Example 1, case b;
FIG. 20. is an example of (PHS) of the polymer clusters modeled in
accordance with Example 2;
FIG. 21. is a graph of projected free volume values versus group
additivity free volumes in accordance with Example 2;
FIG. 22. is a graph of projected free volume values versus the Log
of experimentally obtained Oxygen Permeability in accordance with
Example 2;
FIG. 23. is a graph of projected free volume values versus the Log
of experimentally obtained CO.sub.2 permeability in accordance with
Example 2;
FIG. 24. is a graph of EGDMA weight fraction versus weight
conversion, wherein the EGDMA monomer feed, the instantaneous, and
the cumulative polymer compositions are shown as a function of
total monomer weight conversion;
FIG. 25. is a graph of MBAM weight fraction versus weight
conversion, wherein the MBAM monomer feed and the instantaneous and
cumulative polymer compositions are shown as a function of total
weight conversion;
FIG. 26. is a graph of weight fraction composition for a mixture of
EGDMA and MBAM crosslinkers as a function of conversion in
accordance with Example 3; and
FIG. 27. is a schematic diagram showing the relationship between
the different process steps embodied and broadly described by the
present invention.
THE INVENTION
Applicants have succeeded in developing a process which combines a
summary of molecular mechanics calculations on a variety of stable
minima conformations, in the form of statistical weights or
probabilities, and reaction kinetic expressions. Further, because
the molecular structure is known at all points in the process, it
is possible to estimate the polymer's physical properties by
employing quantitative structure property relationships or more
advanced methods based on the spatial distribution and energetics
of all the atoms in the polymer. Because many properties, including
mechanical and diffusion, are a function of clusters of polymer
molecules instead of isolated molecules, the present process
provides methods whereby the clusters are assembled.
Polymer Chemical Composition Further Composition
In a polymerization system containing two or more monomers, or
monomer types (defined as different conformations of a monomer),
the composition of the polymer being formed usually varies as a
function of conversion. See Skeist, I.; J. of the American Chemical
Society, 68, 1781 (1946) which generalized the work of Alfrey, T.
Jr., and C. C. Price; J. of Polymer Science, 2, 101 (1947) and
Walling and Briggs; J. of the American Chemical Society, 67, 17741
(1945) and formulated general equation to describe the change of
composition as a function of conversion. Skeist's work is based on
the analogy between the variation of polymer composition as a
function of conversion and the Rayleigh equation which describes
the composition of a binary distillation as a function of fraction
distilled. Jean Guillot (Makromol. Chem., Macromol. Symp., 35/36,
269-289 (1990)) uses a different approach, applying reactivity
ratios and empirically-fit kinetic equations to estimate copolymer
bulk compositions.
According to the present invention a method is obtained for
estimating the composition of the polymer as a function of
conversion, then the calculation of numerous physical properties,
which depend on the polymer composition, can be readily performed.
Step (a) of the instant inventive process enables determination of
polymer compositions (monomer sequence distribution) of polymers
containing from two to thirty different monomers or monomer types,
using chemical reaction kinetics theory and experimentally measured
monomer reactivities.
FIG. 27 is a block diagram of the overall system of the present
invention identifying in the blocks 10 to 26 the respective
operation in the steps of the process outlines above. FIG. 27 in
block 10 represents the step (a) of determining the proposed
polymeric substance of molecular chemical compositions through
estimation of individual polymer chain chemical composition. Blocks
11-15 represent step 9b) in determining the polymer's molecular
3-dimensional folding by estimating properties of the chemical
composition. As this step 9b) represented by block 11 may be
performed by a plurality of numerical methods block 12 represents
(B1) the global growth method, block 13 represents (B2) the In Situ
Growth Method, block 14 represents (B3) the Phantom Growth Method
and block 15 represents (B4) the Conditional Phantom Growth Method.
Blocks 16 to 18 represents the step (c) of assembling the
3-dimensional folded molecular composition into a polymeric or
copolymeric cluster. As this step 9c) represented by block 16 may
be performed by a plurality of methods, block 17 represents (C1)
the method using molecular excluded volume constraints determined
by vector geometry. And block 18 represents (C2) the method of
using a step (b) interactively. Blocks 19-26 represent step (D) of
estimating the physical properties of the resulting polymeric or
copolymeric cluster.
As this step D represented by block 1-9 may be performed by a
plurality of methods, blocks 20 to 26 represent such methods. Block
20 represents estimation of the physical properties of the
resulting clusters by quantitative property relationships. Block 21
represents the use of the standard molecular modeling technique of
molecular mechanics. Block 22 represents the use of the molecular
dynamics. Block 23 represents Monte Carlo methods. Block 24 free
energy perturbation methods; block 25, the extended volume
constraint method and block 26 represents other available energy
methods. FIG. 4 is a schematic diagram for the steps of defining or
determining the proposed polymeric of copolymeric substance or
material's molecular chemical composition.
The logic flow chart of FIG. 4 illustrates step A operation. In
this step from the mix of potential monomers there is a selection
of information on a sequence of monomers to form a desired monomer.
Block 27 represents the start of the determination process. Block
28 represents the input information on the monomer mixture.
Thus rectangle 28 represents reading in of the monomer types, the
conditions and kind of polymerization which determine the selection
of the monomer for processing. Rectangle 29 represents the
computation of the reactivity ratios for this polymerization. The
oval 50 represents the addition of the next monomer, or the count
for the next step. Rectangle 51 represents the integration of
chemical equation by the Adams- Moulton step, the output of which
goes to the decision step 52 to determine whether the monomer
conversion has reached the desired polymer. If not, the procedure
loops back to oval 50 and repeats the integration step. If yes, the
procedure then goes to monomer-polymer sequences to a decision
point.
Rectangle 53 represents the selected monomer and the polymer's
chemical composition. The oval 54 represents the conversion and
rectangle 55 represents the calculations on the procedure, followed
by the addition of monomer represented by the following rectangle
57 using the calculations product in selecting the monomer type. At
the decision diamond 58 the question is, has the polymer grown to
he desired length? If no, the procedure loops back to oval 56 to
add a monomer to get the chain long enough, or to the desired
degree of polymerization. If the decision is yes, the rectangle 59
represents the step of storing the complete monomer sequence for
later use as for example in Process Step B.
Diamond 60 represents the decision point for the question, is the
procedure OK, if no it should be repeated for a different level of
conversion i.e. a different kind of polymer. If the decision is no,
the procedure loops back to the addition command at oval 54, before
the Adams-Moulton step. This represents doing the step A more than
once for the selected polymer. Stated otherwise, the question is,
in the monomer sequence is the order of the polymer chain length
adequate? If yes, the rectangle 61 represents proceeding to a
display of the polymer. Diamond 62 represents the decision point on
the question whether to proceed to polymer folding. If yes, then
proceed to Step B, rectangle 11. If not, proceed to stop point
63.
3-Dimensional Polymer Folding
As stated above, small molecules have properties dependent mostly
on the equilibrium structure of just a few conformations. A polymer
has a hugh number of roughly equivalent "stable" structures.
Experimentally it has been found that most polymer structures
follow Gaussian statistics in their spatial distributions. An
existing theory is able to explain such experimental findings, and
mathematical expressions are available which reproduce the
experimental results quite well. (See Flory, P. J., "Spatial
Configuration of Macromolecular Chains", Nobel Lecture, Dec. 11,
1974, in "Selected Works of Paul J. Flory, Mandelken, Mark, Suter,
Yoon Eds. Standford University Press, V. 1. (5), 1985.) The instant
invention utilizes such theory to estimate the shape, by
three-dimensional folding, of large polymer molecules. The methods
for growing polymers from the monomer sequence distribution are
Global Growth, In Situ Growth and Simple and Conditional Phantom
Growth. Each method uses techniques similar to the molecular
statistical thermodynamic theories found in Flory, P. J.,
"Statistical Mechanics of Chain Molecules", Wiley-Interscience, New
York. (1969).
Global Growth Method
This growth method is repeated using of the previously generated
sequence until entire polymeric chain growth is completed as
illustrated by this loop back at decision point 44.
In order for the polymer conformation to be representative of
=Blotzmann distribution of energies at temperature T, global
minimization of a Molecular Mechanics force field is used to
generate statistical weights for each of the possible
conformations. For vinyl polymers without side chains there are 9
possible conformations at each new monomer addition
______________________________________ tt tg+ tg- g + t g + g+ g +
g- g - t g - g+ g - g- ______________________________________
After global minimization of each of the nine structures, the
energy is kept and probabilities are calculated from the minimized
energies of each conformation (j=1,. . . ,9) of the polymer
including the i-th monomer:
where the E.sub.ij 's depend on the conformation of the previous
i-1 monomers, and ##EQU1##
FIG. 5 illustrates the global growth outline of polymer folding.
First we read in the x, y, z coordinates of the monomers in the
polymer sequence. We start the 3-d polymer connectivity with the
first monomer from the previously generated sequence ##EQU2##
This growth method is repeated for each monomer of the previously
generated sequence until the entire polymer chain is grown.
The logic flow chart of FIG. 5 illustrates the operation of the
global Growth Method of the polymer folding. The block 30
represents the initiation of the program which is repeated for each
monomer of the previously generated sequence until the entire
polymer is generated. Block 31 represents allocation of the
acquired x, y, z coordinates of each of the monomers of the
sequence. The molecular structure file has the coordinates in it
from data previously acquired and represents how the sequence is
put together. This precedes data processing. Rectangle 32
represents initializing the data processing by initializing the
molecular connectivity table and a selection of angles. The angles
are represented by the possible conformation referred to above, as
for example 9 for vinyl polymers. Diamond 33 represents the
decision point as to how many angles are desired. The angles are
described above. The rectangles 34 to 43 represent the steps to the
decision point 44 as to whether the polymer is complete. It will be
understood that the program loops back at oval 39 to the angle
recording step at 35, if all the conformations have not been
accomplished. Rectangle 35 represents determining the angle of the
next sequential conformation. Rectangle 36 represents adding the
conformation. Rectangles 37 and 38 represent energy minimization.
Rectangles 40, 41, 42 and 43 represent the determination of
Boltzman distribution of the polymer conformation to provide the
polymer chain.
At the decision point 44 the question is, is the polymer chain
complete? If the answer is not, the procedure loops back thru the
instruction in the oval 45 to add.
If the answer is in the affirmative and the entire sequence is
grown, the procedure goes to step C which is the cluster step
illustrated in FIG. 27, 9 and 10. Step C is provided for occasions
when a cluster step is needed.
In theory this would provide the best answer to the determination
of a Boltzmann distribution of polymer conformations at a given
temperature T. In practice, however, global minimization is too
slow for the size of polymers (i>10) of interest. The
calculation of minimized energy grows at the rate of the square of
the number of atoms in the chain (N.sup.** 2). Therefore,
approximations imply a reduction of the range and type of
correlations between atoms along the polymer backbone. Others
simplify the force field to speed up the calculations.
In Situ Growth Method
One method to mimic global optimization is by In Situ Growth. The
In Situ method considers only the few monomers which are at the
growing end of a chain i.e. the movable monomers. The method
selects the number of monomers to use, usually 3, and a distance
shell around these monomers to include in the surrounding atomic
environment in the calculations. Then a new monomer is added in all
its possible conformations (i.e. for a vinyl monomer, 9 sets of
angles, for a chiral vinyl monomer, 18 sets of angles), each choice
of monomer addition and set of angles describing each conformation
is minimized with the force field, including only atoms in the
chosen environment. The minimized energies are computed, and a
probability vector is generated, with one value for each locally
stable structure. These probabilities are temperature dependent.
Probabilities are calculated from the minimized energies of each
conformation (j=1,. . . ,N) of the polymer including the i-th
monomer following the equation P.sub.ij =exp (-E.sub.ij
/RT)/Z.sub.i.
A random number is generated form a uniform [0,1] distribution and
the structure whose probability interval contains that random
number is chosen. Then the method is repeated. The set of growing
monomers slides down the chain by one monomer unit, and a new
distance environment atom set is selected. All possible
conformations are computed for adding a new monomer, and the
probability vector regenerated from the energies. This approach can
be used to predict solvation effects by including a representable,
yet small, number of solvent molecules within the "sphere" of the
growing polymer end. One way in which this method has been
implemented is by the use of MacroModel's.COPYRGT. substructure
optimization, i.e., energy minimization is constrained to a sphere
(usually a distance equal to 2-10 .ANG.) around the last monomer
added to the chain. The rest of the chain remains fixed although a
few atoms are still allowed to move slightly while being
constrained by parabolic force constants. FIG. 6 is a flow chart
diagram illustrating the In Situ Growth method.
The logic flow chart of FIG. 6 illustrates the operation of the In
Situ Growth Method of polymer folding. The program is initiated by
the reading out of the monomer from the Molecular Structure file.
The rectangle 64 represents allocation of the x, y, z coordinates
of each monomer. Rectangle 65 represents initializing the date
processing of the molecular Connectivity Table and selection of
angles. The diamond 66 represents selection of the monomers and the
angles. The angles are described above. The rectangles 67 to 80
represent steps to the decision as to whether the polymer is
complete. It will be understood that the program loops back at oval
74 to the angle read-in step at 68, if further conformations are to
be added. Rectangle 68 represents determining the angle and
rectangle 69 represents adding the monomer in all possible
conformations. Rectangle 71 represents the minimization of each
conformation. Minimization refers to energy minimization as known
in rectangle 73 represents the decision point as to whether all the
angle sets for the monomer have been completed. The negative branch
from the decision at 73 carries the program back to reading in
another angle set about adding the monomer. The affirmative branch
carries the monomer to the step represented by rectangle 75 and the
Boltzman determination wherein the probabilities are calculated
from the minimized energies of each conformation of the polymer.
Rectangle 76 represents the choosing of a random number described
above, the selection a structure is represented in rectangle 77. In
the next step, represented by rectangle 78 the set of growing
monomers take place and at the decision point of rectangle 80 the
decision question is, is the polymer chain complete? The negative
branch carries the program back to the step of the reading in
another angle set at rectangle 68 toward completing the polymer.
The affirmative branch carries the procedure to step C.
Simple Phantom Growth Method
The Simple Phantom Growth method reduces J. P. Flory's theory to a
Simple Markov process, wherein conformational probabilities depend
only on the conformation of the previously added monomer. As such,
nine probabilities of adding a monomer are calculated from energies
of adding a single monomer to another. Long range effects are
neglected.
Implementation includes intradyad and interdyad probabilities. As
an example consider the following polymer conformation
sequence:
__________________________________________________________________________
(1) Polymer t g+ g+ t t t g- g- g+ t t g+ Conformation (2)
Intradyad (tg+) (g + t) (tt) (g - g-) (g + t) (tg+) Conformations
(3) Interdyad (g + g+) (tt) (tg-) (g - g+) (tt) Conformations
__________________________________________________________________________
Although this method is the fastest way to model the folding of
polymer molecules, the shortness of range of correlations in
neighboring polymer conformations result in highly coiled, and
self-crossing (quite often one self-crossing every four monomers)
chains. Consequently the energy distribution is highly
non-Boltzmann. FIG. 7 is a block diagram illustrating the Simple
Phantom Growth method. Use of the Conditional Phantom Growth method
greatly alleviates the non-Boltzmann distribution.
The logic flow chart of FIG. 7 illustrates the operation of the
Phantom Growth Method of polymer folding. The block 14 represents
the initiation of the program. Rectangle 81 represents reading the
monomer structures form a file. Rectangle 82 represents creating
all possible dimers. For a sample vinyl polymer with no side
chains, this is nine dimers. Rectangle 84 represents minimize the
dimers and saving the optimized energies (.SIGMA..sub.ij) in a
table. Diamond 83 represents the question are the dimers energies
optimized as read in from rectangle 82. The negative branch carries
the program to rectangle 84 and the affirmative branch to rectangle
85 representing an input temperature. Rectangle 86 represents
computing the probabilities by using Boltzman probabilities for the
ith .PHI. and the jth .PSI. and the kth monomer as shown in the
equation ##EQU3##
Oval 87 represents the monomer number and rectangle 88 represents
setting up the monomer connectivity table. The monomer is from an
earlier sequence.
Oval 89 represents adding a monomer with a particular .PHI. and
.PSI. value. Rectangle 90 represents retrieving coordinates of the
mth monomer and selection of a uniform random number. Rectangle 91
represents the value of the random number is compared to the .PHI.,
.PSI. probability intervals in the table. It is noted that the
interval that corresponds to the chosen random number determines
the .PHI., .PSI. value for the monomer addition. Diamond 92
represents checking to determine whether the chain is complete,
i.e. the degree of polymerization has been reached. The negative
branch carries the program back to oval 89 to add another
monomer.
The cycle is repeated until the required different chains have been
generated. The diamond 93 represents checking on completion of the
desired polymers. Rectangle 94 represents proceeding to display of
the polymer. Diamond 95 represents the decision point on the
question whether to proceed to polymer assembly. If yes, then
proceed to Step C, rectangle 16. If no, proceed to stop point
96.
Conditional Phantom Growth Method
In the Conditional Phantom Growth method, longer correlations are
built by computing conformational probabilities from optimized
trimers which include up to five bond correlations. Optimized
"trimer" energies are used to calculated row vectors of conditional
probabilities as follows:
__________________________________________________________________________
Example: Monomer i-1 Monomer i.fwdarw. .dwnarw. g + g g + g- g + t
g - g+ g - g- g - t t g+ t g- t t
__________________________________________________________________________
g + g+ 0.701 0.032 0.000 0.004 0.000 0.000 0.245 0.016 0.002 g + g-
0.000 0.000 0.000 0.829 0.084 0.036 0.049 0.002 0.000 g + t 0.233
0.001 0.000 0.625 0.020 0.016 0.099 0.005 0.001 g - g+ 0.818 0.012
0.000 0.000 0.000 0.000 0.155 0.008 0.006 g - g- 0.000 0.000 0.000
0.854 0.051 0.030 0.061 0.004 0.000 g - t 0.267 0.004 0.001 0.215
0.013 0.009 0.469 0.017 0.004 t g+ 0.631 0.182 0.000 0.000 0.000
0.000 0.170 0.016 0.001 t g- 0.000 0.000 0.000 0.295 0.032 0.027
0.621 0.023 0.003 t t 0.162 0.002 0.000 0.550 0.026 0.029 0.223
0.008 0.000
__________________________________________________________________________
Once a conformation is selected for the Current monomer, a van der
Waals distance check is conducted. If large van der Waals
repulsions are found then one monomer is backtracked and a
different conformation is chosen (in the average the second most
probable conformation will be chosen). If no conformation can be
found to overcome van der Waals repulsion, the In Situ Growth
method is used for optimization of the current monomer
addition.
Conditional Phantom Growth provides to the average probability
weights which are good approximations to the average values
obtained in the In Situ method. The folding of polymer molecules is
quite efficient, hundreds of folded polymers can be calculated in a
matter of minutes. The resulting energy distribution closely
follows Boltzmann statistics, there is some long range conformation
correlation and molecular energies are very often acceptable even
though the initial monomer geometries used have only nominal values
for t, g+, and g- angles. Quite often the molecular structures can
be minimized without singularities when no self-crossing occurs.
The backbone conformation preserves its general shape even after
minimization and end-to-end distances vary by less than 1% after
minimization. FIG. 8 is a block diagram illustrating the
Conditional Phantom Growth method.
The logic flow chart of FIG. 8 illustrates the operation of the
Conditional Phantom Growth method. The rectangles, diamonds and
ovals of FIG. 8 represents similar steps in the flow chart of FIG.
7 which is referred to. A difference is found at rectangle 100 in
FIG. 8 which represents computing conditional probabilities for
adding monomer type k in the h-th conformation given that the last
monomer added was of type j in the 1-th conformation.
Polymer Cluster Assembly
The assembly of the resulting 3-dimensionally folded molecular
composition into a proposed polymeric or copolymeric cluster is
created as follows. From a statistical distribution of individual
molecules, a polymer cluster reminiscent of the polymer in the bulk
is fabricated. Then the model builder selects one of two methods;
one method uses new excluded volume constraints derived from
analytical vector geometry and the individual polymer structures of
step (b). See figure (9), from "Molecular Silverware: General
Solutions to Excluded Volume Constrained Problems", by Mario
Blanco, Journal of Computational Chemistry, accepted for
publication Jul. 3, 1990.
For the purpose of resolving molecular overlaps each molecule is
modeled as a rigid body, i.e., as a collection of hard spheres. The
size of each sphere is set by the van der Waals radius of the
corresponding atom. Of course, covalently bound atoms necessarily
have overlapping van der Walls spheres. We are only concerned with
the resolution of van der Walls overlaps between nonbonded
atoms.
Two rigid bodies require six degrees of freedom to have their
relative orientations and positions in free space completely
specified. We choose the first three degrees of freedom to be the
euler angles, .OMEGA.=(.alpha.,.beta., .gamma.), which define the
orientation of molecular 2 in a coordinate frame affixed to
molecule 1. The vector connecting the centers of mass of both
molecules, in polar coordinates (r, 0, .phi.), completes the list
of six, initially independent degrees of freedom. One of these r,
becomes a function of the other five when the excluded volume
constraint equations are enforced. We seek the form and the
solution of these equations.
The global coordinate reference frame is depicted in Annex FIG. 1.
The frame is affixed to the center of mass and oriented along the
principal moments of inertia axes of molecule 1. The atoms are
number i=1,2, . . . ,N.sub.1. Atoms in molecule 2 are numbered
j=N.sub.1 +1,N.sub.1 +2, . . . ,N.sub.1 +N.sub.2. In the final
self-avoiding assembly the vector location of all atoms in molecule
2 will be given by two rigid body operations, a rotation
R.sub..OMEGA. (.alpha., .beta., .gamma.) and a translation r n,
applied to the original atomic position vectors r.sub.j.sup.O
n is a unit vector in the polar direction (0, .phi.),
The excluded volume constraint equations can be written in the
following general form
s.sub.i and s.sub.j are the van der Walls radii of the
corresponding atoms. Because in the first case "molecule" 2 is
monoatomic the Euler angles .alpha., .beta., and .gamma. are all
set to zero and the rotation matrix becomes a 3.times.3 unit
matrix.
Annex FIG. 2 depicts how the excluded volume constraint expression
(5) was obtained. An equation for the magnitude r of the
translation vector along the n line of sight was derived from the
expression for the magnitude of the vector addition
Expansion of the left hand side leads to expression (5). The
solution and a physical interpretation of expression (5)
follows.
The displacement needed to place the atom j in contact with atom i
along a line of sight given by the direction n is obtained by
setting .PHI..sub.ij to the minimum value allowed by the
contstraint eq. (4), i.e., .PHI..sub.ij (r, 0, .phi.)=0. Solving
for r one gets ##EQU6##
r.sub.ij places atom j on the near side of atom i while
r.sub.ij.sup.+ places it on the far side. Negative values of
.sub.ij r.sub.- or
r.sub.ij.sup.+ are interpreted as positive displacements along the
-n direction. Notice that two complex roots result if
Complex roots indicate that no van der Walls overlap exists along
the line of sight n and thus no real displacement can put atom i
and j in contact with each other. Annex FIG. 3 depicts all six
possible root combinations in terms of the initial positions of
atoms i and j and the n line of sight.
A more useful interpretation of eq. (4) follows. The rear roots of
.PHI..sub.ij (r, 0, .phi.) define a set of open segments on the
real axis
where the index q runs only over the p real roots of .PHI..sub.ij
(r, 0, .phi.). If atom j gets translated through a vector .PHI.n,
with .PHI. a number contained within any .lambda..sub.qj segment,
##EQU8## it violates constraint eq. (4). The union of all
.lambda..sup.qj open segments constitutes the complete set of
"forbidden" displacements. Conversely, the complement of the union
defines all points along the line of sign n where atom j can be
translated and be free from van der Walls overlaps with molecule 1.
Therefore, the solution to the excluded volume constraint eq. (4)
is the set ##EQU9##
FIG. 4 illustrates the effect of the union operation on the
possible placement locations for atom j when all van der Waals
overlaps are resolved. Note that because the .lambda..sub.qj 's are
open segments, their ends, the real roots of .PHI..sub.ij (r, 0,
.phi.), may be part of the set .lambda..sub.j.sup.c. However, the
union operation eliminates most of the roots from the solution set.
It is relatively simple to write computer code to determine which
roots survive the union operation. Two roots which are never
eliminated are ##EQU10##
These two immortal roots are always elements of the solution set
.lambda..sub.j.sup.c because there are no segments to the right and
to the left of the right-most and left-most segments respectively.
All displacement values in the ranges ##EQU11## are also elements
of the solution set .lambda..sub.j.sup.c. Incidentally if one
leaves all values greater than r.sub.j.sup.+ and less than
r.sub.j.sup.- out of the solution set .lambda..sub.j.sup.c one
obtains an operational definition of the "probe" accessible
surface. If the van der Waals radius s.sub.j is set equal to zero
one get an operational definition of the multiple valued van der
Walls surface of molecule 1 in polar coordinates.
The Euler angles are required in this case. There are N.sub.1
.times.N.sub.2 constraint equations similar to (4) ##EQU12## all
other quantities have their previous meanings. The general solution
to the excluded volume constraint problem is
For simplicity of notation all roots of eq. (13) were made to
appear in the definition of solution (15) but it must be kept in
mind that only the real roots are to be included. Just as before we
have ##EQU13## Expression (15) makes no assumptions regarding
molecular shape. All valid excluded volume constraint solutions can
be found in the .lambda..sup.c set. All physically allowed
self-avoiding assemblies can be created regardless of the
complexity in the topologies of molecules 1 and 2 by changing the
values of (.alpha., .beta., .gamma.) and 0, .phi.) and solving for
the .lambda..sup.c set following eqs. (16)-(18).
The operational simplicity of solution (15), its generality with
respect to molecular topology, and the ability to select
appropriate paths through (.alpha., .beta., .gamma.) and (0, .phi.)
space are important elements for the search and development of more
general molecular modeling algorithms appropriate to condensed
phase simulations. From such a molecular modeling point of view an
important element of the solution set is the value r.sup.3 defined
such that
The vertical bars indicate absolute values. r.sup.c is the shortest
possible displacement required by molecule 2 to resolve all atomic
overlaps with molecule 1.
The excluded volume constraint problem can be stated in the most
general possible way as follows.
Given M molecules with fixed orientations and center of mass
directions
respectively, find the complete set of all molecular displacements
(r.sub.m 's) which satisfy the excluded volume constraint
equations: ##EQU14## where r.sub.j.sup.o and r.sub.j.sup.o are the
original position vectors for atom i and atom j respectively. m'
and m identify the molecules carrying atoms i and j respectively,
and N.sub.m (m=1,2, . . . ,M) gives the number of atoms in molecule
m.
The elements of the solution set .lambda..sup.c are M-tuples of the
form (r.sub.1, r.sub.2, . . . ,r.sub.M). There is an infinite
number of elements in .lambda..sup.c. Most of them belong inside
semi-infinite segments of the form (12c). From a molecular modeling
point of view, however, the important solutions satisfy eq. (19)
also.
Equation (20) defines a nonholonomic constraint problem. There are
no general ways of attacking nonholonomic constraint problems.
According to Goldstein.sup.5 "the most vicious cases of
nonholonomic constraints must be tackled individually." A practical
solution to this problem is obtain by extending the method employed
in the M=2 case, i.e., the components of the M-tuples are found one
at a time.
We begin by placing one of the M molecules at the origin of the
global reference frame (r.sub.1 =0). We then selected a second
molecule and apply the method of the previous section to find the
solution set for the two molecule system. In accordance with eq.
(19) we set r.sub.2 equal to min .vertline..lambda..sup.c
.vertline.. Notice that we could have selected the identity of the
first and second molecules in M and M-1 different ways
respectively. In general there are M-m+1 ways of choosing the
M.sup.th molecule when the assembly already contains m-1 molecules.
Consequently there is a possible total of M! different molecular
assemblies. Each of these assemblies is located fully in 3-D space
by giving (1) the original coordinates of the atoms in each
molecule r.sub.j.sup.o. (2) the orientations and displacement
directions, .OMEGA..sub.m and n.sub.m respectively, and (3) the
M-tuple (r.sub.1,r.sub.2, . . . ,r.sub.M) of molecular centers of
mass displacements.
The solution method consists of the following steps
1. Choose a labeling scheme for the M molecules
2. Define the following operational indexes: ##EQU15##
3. Set the global coordinate frame to coincide with the center of
mass of molecule 1. In effect we are setting r.sub.1 =0. Set
m=1.
4. Calculate the value r.sub.m=1 according to the solution method
employed in the M=2 case, i.e., .vertline.r.sub.m+a
.vertline.=Min.vertline..lambda..sup.c .vertline.. Equations (14)
and (18) now read ##EQU16## Reset the coordinates of the m+1
molecule according to ##EQU17## set M to m+1.
6. Repeat steps 4 and 5 until m=M-1.
The other uses the growth method of step (b) iteratively to create
a polymer assembly.
The logic flow chart of FIG. 9 is a diagram of the excluded volume
polymer assembly method. Rectangle 97 represents reading in from
the polymer file which has a number of single polymer chains in the
file. Rectangle 98 represents arbitrary choosing how to read the
polymer chains in from the file using either a random number or to
select a particular chain from the file. Rectangle 99 represents
setting the origin of the system as the center of mass of the first
chain. The oval 101 represents the selection of a second polymer
from the determination. Rectangle 102 represents orientation of the
polymer in all possible ways to eliminate all overlaps of atoms in
the two structures and the two structures are as close as possible
to each other without overlapping.
It is noted that there is no change in the internal angles of the
polymer chain, but only rotation and twisting of the entire chain
to avoid overlaps. This is computed using the equation of the
article entitled "Molecular Silverware" (J. Computational
Chemistry, Vol. 12, No. 2 pp. 237-247 (1991).
Rectangle 103 represents growing of the polymer cluster by adding
the current molecule displaced from the center of the mass in the
preceding step. Diamond 104 represents checking to determine
whether the full assembly has been created. If yes, proceed to
display of rectangle 105. If not, the program returns to oval
101.
From display at 105 the program proceeds to the question in diamond
106, are properties to be estimated. If yes, program proceeds to
Step D, property estimation and analysis, block 19. The negative
branch goes to diamond 107 for decision on the question, of
choosing a different labeling scheme. An affirmative branches back
to rectangle 99 and a negative to stop point 108. FIG. 10 is a
block diagram illustrating the multi-chain growth method.
Estimation of the physical properties of the resulting clusters is
achieved through the use of molecular energy expressions and/or
quantitative structure property relationships, applied to the
outcomes of steps (a), (b), and (c). Physical property estimation
may also be achieved by any of the standard molecular modeling
techniques, including but not limited to molecular mechanics,
molecular dynamics, Monte Carlo methods, free energy perturbation
methods, or the analytical vector geometry method described herein.
Examples of properties computed are:
Thermal Properties: Glass Transition Temperature
Optical Properties: Refractive Index
Diffusion Properties: Specific Free Volume CO.sub.2, O.sub.2
Diffusion Coefficients
Mechanical Properties: Elastic Moduli
The logic flow chart of FIG. 10 illustrates the operation of a
Multichain-Growth Polymer Assembly Method, which uses the growth
method to create an assembled polymer structure. Rectangle 109
represents selecting how large assembly should be, i.e. the number
of chains. Rectangle 110 represents generating starting points over
the surface of a sphere. Rectangle 111 represents the selection of
growth method. For each chain one of the starting points is
selected as represented by the rectangle 112. And a chain is grown
into the sphere employing a suitable method in accordance with the
above description.
The rectangle 113 represents the question, are the chains all
finished? If yes, the program proceeds to Step D, property
estimation and analysis. If no, the program lops back to oval 114
and an addition of a chain.
The process proceeds until all the starting points are used and the
desired chains grown. At that point a polymer is completed.
The following Examples serve to demonstrate aspects of the present
inventive process.
EXAMPLE 1
Estimation of chemical molecular composition of a polymer reacted
from a monomer mixture of methyl acrylate (MA) and styrene
(STY)
The chemical composition of a copolymer rarely remains constant
both in its overall instantaneous monomer percent concentration and
monomer sequence distribution, during the course of a
polymerization reaction. The physical properties of the copolymer
are highly dependent upon such chemical drifts. It is for this
reason that the ability to estimate the molecular composition of a
copolymer at different points during the reaction (percent
conversion) is so important.
As an example, consider the copolymerization of methyl acrylate
with styrene. The present invention used published chemical
kinetics experimental information (in the form of Q and e values)
to estimate the composition of the copolymer as a function of
conversion. Two cases are presented:
Case a: The starting composition is a 50:50 mixture of MA and
STY
Case b: The starting composition is a 23:77 mixture of MA and
STY
______________________________________ Case a: Input Information
______________________________________ L (Lin/Pha/CPha) 2 (# unique
mons) 50 (chain length) MA (Monomer 1 Name) 1.0000 0.420 0.650
86.09 8.0 1.4800 1 (Conc Q E MW TG RI Frag) STY 1.0000 1.000 -0.800
104.14 100.0 1.5900 31 (Conc Q E MW TG RI Frag) M (Mole or Weight
Basis) -987765440 (Random # seed)
______________________________________
The first line refers to the type of polymer chain to be generated,
i.e. In Situ, linear, Phantom, Conditional Phantom. Polymer
examples having a length of 50 monomers were requested. FIG. 11 ia
a computer generated picture of the instantaneous polymer
composition as a function of monomer conversion. The conversion
values are 0.0%, 25%, 50%, 75% and 100% from top to bottom
respectively. Note the small drift in composition due to the
differences in reactivities between the two monomers. This agrees
with the experimental preparation of this polymer.
Using the In Situ methods, three dimensional structures of these
copolymers were generated from Case a. FIGS. 12-17 show the
structural variations of these differing sequences copolymer chains
at the 0%, 20%, 40%, 60% 80% and 99% monomer conversion levels.
With a 50:50 mixture of starting material, a very different set of
structures were achieved during the course of a batch reaction.
This leads to a wide variation in polymer shapes and
compositions.
At the end of the reaction (close to 100% conversion), the
copolymer that is being prepared is almost exclusively made up of
methyl acrylate. Styrene has a larger preference for reacting with
itself and it forms strings of connected styrene monomers even from
the start of the reaction. At close to 100% conversion, most of the
styrene has already been incorporated into the reacted polymer
leaving only methyl acrylate to form polymer structures (FIG.
17).
__________________________________________________________________________
Output Information ***Conversion-Composition (Mole Fraction
Basis)*** # Monomer Name Q E MWT TG RI
__________________________________________________________________________
1: MA 0.420 0.650 86.09 8.0 1.4800 2: STY 1.000 -0.800 104.14 100.0
1.5900
__________________________________________________________________________
Monomer Instantaneous Cumulative Conversion Mix Polymer Polymer
__________________________________________________________________________
0.000 Monomer 1 0.5000 0.3798 0.3798 Monomer 2 0.5000 0.6202 0.6202
Glass Trans Temp (Deg C.) 63.0 63.0 Refractive Index 1.5530 1.5530
0.200 Monomer 1 0.5283 0.3944 0.3867 Monomer 2 0.4717 0.6056 0.6133
Glass Trans Temp (Deg C.) 61.7 62.4 Refractive Index 1.5515 1.5523
0.400 Monomer 1 0.5697 0.4158 0.3955 Monomer 2 0.4303 0.5842 0.6045
Glass Trans Temp (Deg C.) 59.7 61.6 Refractive Index 1.5493 1.5514
0.600 Monomer 1 0.6384 0.4526 0.4077 Monomer 2 0.3616 0.5474 0.5923
Glass Trans Temp (Deg C.) 56.2 60.4 Refractive Index 1.5453 1.5501
0.800 Monomer 1 0.7862 0.5502 0.4285 Monomer 2 0.2138 0.4498 0.5715
Glass Trans Temp (Deg C.) 47.3 58.5 Refractive Index 1.5347 1.5479
0.990 Monomer 1 1.0000 1.0000 0.4949 Monomer 2 0.0000 0.0000 0.5051
Glass Trans Temp (Deg C.) 8.0 52.3 Refractive Index 1.4800 1.5408
__________________________________________________________________________
Case b: Input Information
__________________________________________________________________________
L (Lin/Pha/CPha) 2 (# unique mons) 50 (chain length) MA (Monomer 1
Name) 0.233 0.420 0.650 86.09 8.0 1.4800 1 (Conc Q E MW TG R STY
(Monomrt 2 Name) 0.767 1.000 -0.800 104.14 100.0 1.5900 31 (Conc Q
E MW TG R M (Mole or Weight B -987765440 (Random # seed)
__________________________________________________________________________
Only the initial monomer concentrations have been changed (methyl
acrylate 0.233, styrene 0.767). Note that the drift in composition
is drastically reduced when the initial monomer feed composition is
changed from 50%:50% to 23.3%:76.7%. In fact, this composition is
called the "azeotropic" composition. An azeotropic composition is
that which balances the differences in reactivities of the two
monomers by setting their initial concentrations such that at all
times during the reaction the copolymer composition remains
constant.
FIG. 18 shows the copolymer generated from example 1.b. In this
case the polymer sequences and structures are the same throughout
most of the reaction. Consequently, FIG. 18 is a polymer structure
generated at all of the conversion levels 20%, 40%, 60% and 80%.
The structure slightly changes at 99% and is shown in FIG. 19. The
identical sequences and their similar three dimensional shapes
create uniform polymer shapes and compositions during the course of
the batch reaction.
The present invention allows for the location of such azeotropic
compositions over all or just part of the polymer reaction. This
invention also shows the differing polymer shapes and consequently
describes many of their properties such as size, volume, stiffness,
and polymer structure. From these structures many other polymer
design problems may be considered, such as packing, tensile
strength, glass transition temperature, and other polymer
properties which rely on the inherent three dimensional shape of
polymers.
__________________________________________________________________________
Output Information ***Conversion-Composition (Mole Fraction
Basis)*** # Monomer Name Q E MWT TG RI
__________________________________________________________________________
1: MA 0.420 0.650 86.09 8.0 1.4800 2: STY 1.000 -0.800 104.14 100.0
1.5900
__________________________________________________________________________
Monomer Instantaneous Cumulative Conversion Mix Polymer Polymer
__________________________________________________________________________
0.000 Monomer 1 0.2330 0.2243 0.2243 Monomer 2 0.7670 0.7757 0.7757
Glass Trans Temp (Deg C.) 77.8 77.8 Refractive Index 1.5688 1.5688
0.200 Monomer 1 0.2350 0.2257 0.2250 Monomer 2 0.7650 0.7743 0.7750
Glass Trans Temp (Deg C.) 77.7 77.8 Refractive Index 1.5686 1.5687
0.400 Monomer 1 0.2378 0.2277 0.2258 Monomer 2 0.7622 0.7723 0.7742
Glass Trans Temp (Deg C.) 77.5 77.7 Refractive Index 1.5684 1.5686
0.600 Monomer 1 0.2422 0.2307 0.2269 Monomer 2 0.7578 0.7693 0.7731
Glass Trans Temp (Deg C.) 77.2 77.6 Refractive Index 1.5681 1.5685
0.800 Monomer 1 0.2510 0.2368 0.2285 Monomer 2 0.7490 0.7632 0.7715
Glass Trans Temp (Deg C.) 76.6 77.4 Refractive Index 1.5675 1.5684
0.990 Monomer 1 0.3253 0.2843 0.2321 Monomer 2 0.6747 0.7157 0.7679
Glass Trans Temp (Deg C.) 72.1 77.1 Refractive Index 1.5628 1.5680
__________________________________________________________________________
EXAMPLE 2
Estimation of the Gas Permeation Properties of Packaging Material
Containing Para-Hydroxy Styrene (PHS)
A series of para-substituted polystryenes investigated to study the
effect of chemical composition of free volume (Table I).
TABLE I
__________________________________________________________________________
Modeled Substituted Polystyrenes Calculated Density Free Volume
POLYMER X (gr/cc) (cc/gr)
__________________________________________________________________________
Poly(p-chlorostyrene) PCS --Cl 1.246 0.262 Poly(p-hydroxystyrene)
PHS --OH 1.173 0.274 Poly(p-fluorostyrene) PFS --F 1.176 0.292
Poly(p-methoxystyrene) PMxS --OCH3 1.118 0.295
Poly(p-acetoxystyrene) PAS --OCOCH3 1.168 0.297
Poly(alpha-methylstyrene) PaMS --H (a) 1.065 0.299 Polystyrene PS
--H 1.048 0.325 Poly(p-methylstyrene) PMS --CH3 1.009 0.346
Poly(p-t-butylstyrene) PtBS --C(CH3)3 0.947 0.396
__________________________________________________________________________
For each of these polymer, the process of the instant invention was
implemented. The degree of packing in the assembled cluster
estimated in Step (d) also utilized the following expression:
where "d" is the polymer's experimental density, "No" Avogadro's
number, "Mc" and "Vc" the calculated mass and molar volume occupied
by the assembly of polymer molecules. FIG. 20 shows an example
(PHS) of the polymer clusters modeled with the invention and used
to calculate the quantities Mc and Vc in formula (3).
Table I shows the calculated void volume values for the family of
para-substituted polystyrenes. FIG. 21 shows a comparison of the
instant application's projected free volumes with those obtained
using the experimentally based method commonly referred in the
literature as "group additivity". Dirk W. van Krevelen, "Properties
of Polymers, Their Estimation and Correlation with Chemical
Structure", 2nd edition, Elsevier Science Publishers B.V., 1987.
Applicants' predictions and the group additivity results are in
good qualitative agreement. From the results one would conclude
that among the para-substituted polystyrenes PHS has a low
permeability because its void volume is among the lowest of the
para-substituted styrenes while PtBS should have one of the highest
permeabilities by virtue of its high free volume value.
The predictions of the modeling of the instant invention were then
compared against experimental data. The system of para-substituted
polystyrenes has since been studied by Prof. Donald R. Paul, at the
University of Texas at Austin. Puleo A. C., Muruganandam N., Paul,
D. R. "Gas Sorption and Transport in Substituted Polystyrenes",
Journal of Polymer Science Part B-Polymer Physics, 1989, V27, N11,
pp. 2385-2406. Comparison was based upon a well known expression
relating gas permeability and free volume values:
Taking logarithms on both sides of this expressions one should
obtain straight lines when plotting Log P vs Log (1/Vf). The lines
were fitted with a simple expression of the form:
where P.sub.o and V.sub.o are constants which depend only on the
type of diffusing gas, e.g. oxygen or carbon dioxide. The
experimental vs. projected values are shown in Tables II.A and II.B
for oxygen and carbon dioxide respectively. The Experimental values
are obtained from the Paul article. Table III contains the values
of the constants in the permeability model.
TABLE II.A ______________________________________ Experimental vs.
Calculated (Eq. 4) Oxygen Permeability Values PO.sub.2.sup.(1)
POLYMER X Exp. Pred. ______________________________________
Poly(p-chlorostyrene) PCS --Cl 1.20 0.4357 Poly(p-hydroxystyrene)
PHS --OH 0.12 0.7424 Poly(p-fluorostyrene) PFS --F 4.40 1.5214
Poly(p-methoxystyrene) PMxS --OCH3 2.60 1.7001
Poly(p-acetoxystyrene) PAS --OCOCH3 3.10 1.8284 Poly(alpha-methyl-
PaMS --H (a) 0.82 1.9646 styrene) Polystyrene PS --H 2.90 4.6111
Poly(p-methylstyrene) PMS --CH3 7.20 8.3645 Poly(p-t-butylstyrene)
PtBS --C(CH3)3 35.50 26.7805 ______________________________________
.sup.(1) 10**10 [cc(STP)cm/cm**2 sec cm Hg
TABLE II.B ______________________________________ Experimental vs.
Calculated (Eq. 4) Carbon Dioxide Permeability Values
PO.sub.2.sup.(1) POLYMER X Exp. Pred.
______________________________________ Poly(p-chlorostyrene) PCS
--Cl 4.3 3.7303 Poly(p-hydroxystyrene) PHS --OH -- (2) 5.6067
Poly(p-fluorostyrene) PFS --F 17.2 9.7025 Poly(p-methoxystyrene)
PMxS --OCH3 18.9 10.5622 Poly(p-acetoxystyrene) PAS --OCOCH3 16.3
11.1666 Poly(alpha-methyl- PaMS --H (a) 3.0 11.7968 styrene)
Polystyrene PS --H 12.4 22.6472 Poly(p-methylstyrene) PMS --CH3
29.8 35.7041 Poly(p-t-butylstyrene) PtBS --C(CH3)3 140.1 86.9049
______________________________________ .sup.(1) 10**10
[cc(STP)cm/cm**2 sec cm (2) not measured by D. Paul
TABLE III ______________________________________ Gas Permeability
Model Parameters GAS Po (1) Vo (2)
______________________________________ Oxygen O2 11.340 3.188
Carbon Dioxide CO2 10.620 2.437
______________________________________ (1) 10**10 [cc(STP)cm/cm**2
sec cm Hg) (2) cc/gr
FIG. 22 illustrates graphically the qualitative agreement between
the model's predictions (Vf) and the experimental permeability
results. Similar results are obtained for carbon dioxide
permeability values (FIG. 23). The fits indicate that the
calculated Vf values are in good qualitative agreement with the
experimentally measured gas permeabilities. Note in particular that
the projections concerning the permeabilities of PHS and PtBS were
confirmed by Prof. D. Paul's studies. PHS gives the lowest oxygen
permeability values across the family of substituted styrenes while
PtBS gives the highest value. Furthermore, it is clear that the low
permeability of PHS is due greatly to a highly reduced polymer void
volume, a quantity that is calculated using the polymer modeling
methods of the instant invention.
EXAMPLE 3
Copolymerization Kinetics of Acrylic Acid (AA) with Sodium Acrylate
(NaA) Using N,N'-Methylenebis(Acrylamide) (MBAM) and Ethylene
Glycol Dimethacrylate (EGDMA) as Crosslinkers
This Example demonstrates the use of the present invention to
determine a suitable crosslinker for the copolymerization of
Acrylic Acid with Sodium Acrylate. One wishes the crosslinker to be
incorporated into the copolymer in a more or less uniform way
throughout the reaction. Table II shows the estimated compositional
drifts that should be observed with EGDMA as the crosslinker with a
starting composition of
by weight. FIG. 24 is a graph of EGDMA weight fraction versus
weight conversion. The EGDMA monomer feed and the instantaneous and
cumulative polymer compositions are shown as a function of total
monomer weight conversion. Note that the instantaneous copolymer
composition at 90% conversion suffers an 8% depletion in EGDMA
compared to the starting composition. Depletion of EGDMA continues
and reaches 40% at 99% conversion.
Crosslinker compositional drifts may be undesirable in such
instances. The MBAM crosslinker is a better choice if one wishes
the crosslinker to become incorporated into the polymer at a more
uniform rate. Table III shows that for the same polymerization
reaction the MBAM crosslinker drifts by less than 26% during the
entire reaction. Based upon this chemical composition estimate, one
would conclude that MBAM is a better choice of crosslinking
agent.
More detailed compositional information can also be estimated with
the present invention. Table IV shows the chemical sequence
distribution (Dyads, Triads, and monotonic sequences of length up
to 20) for the AA/NaA/MBAM terpolymer as a function of conversion.
Notice that the MBAM crosslinker (C) reacts five times more often
with Sodium Acrylate (B) than with Acrylic Acid (A). For example,
at 90% conversion the dyad distributions are AC+CA=0.171% and
BC+CB=0.935%. This is not just the effect of the large
concentration of Sodium Acrylate in the starting composition (77%)
but also the result of the differences in the binary reactivity
ratios. Table V shows these values. R1 is the ratio of reactivity
of monomer 1 toward itself to the reactivity of monomer 1 toward
monomer 2 (R1=k11/k12, R2=k22/k21). Thus, MBAM is 1.5 times
(R2'/R2") more reactive toward Sodium Acrylate than toward Acrylic
Acid.
Finally, one may choose a balanced mixture of these two
crosslinkers to obtain a more uniform incorporation of crosslinker
throughout the reaction. FIG. 26 shows the instantaneous polymer
composition for a mixture consisting of 0.55% MBAM and 0.45% EGDMA.
Note that because the individual crosslinking agents have opposite
trends in their reactivities as a function of conversion, the
selected mixture comes very close to the desired uniform
incorporation of crosslinking agent throughout the entire
conversion range.
TABLE II
__________________________________________________________________________
Ternary Copolymer Kinetics for AA/NaA/EGDMA
__________________________________________________________________________
Starting Composition: 22/77/1:AA/NaA/EGDMA Molecular MONOMER NAME Q
E Weight
__________________________________________________________________________
MONO 1 AA 1.1500 0.7700 72.06 MONO 2 NAA 0.7100 -0.1200 94.05 MONO
3 EGDMA 0.8800 0.2400 198.22
__________________________________________________________________________
OUTPUT UNITS = WEIGHT FRACTIONS MONOMER INSTANTANEOUS CUMULATIVE
CONVERSION MIX POLYMER POLYMER
__________________________________________________________________________
0.000 MONOMER 1 0.2200 0.2835 0.2835 MONOMER 2 0.7700 0.7060 0.7060
MONOMER 3 0.0100 0.0106 0.0106 0.100 MONOMER 1 0.2133 0.2769 0.2803
MONOMER 2 0.7768 0.7126 0.7092 MONOMER 3 0.0099 0.0105 0.0106 0.200
MONOMER 1 0.2058 0.2694 0.2768 MONOMER 2 0.7843 0.7201 0.7127
MONOMER 3 0.0099 0.0105 0.0105 0.300 MONOMER 1 0.1973 0.2607 0.2729
MONOMER 2 0.7929 0.7288 0.7166 MONOMER 3 0.0098 0.0105 0.0105 0.400
MONOMER 1 0.1876 0.2507 0.2686 MONOMER 2 0.8028 0.7389 0.7209
MONOMER 3 0.0097 0.0104 0.0105 0.500 MONOMER 1 0.1761 0.2386 0.2639
MONOMER 2 0.8144 0.7511 0.7256 MONOMER 3 0.0095 0.0103 0.0105 0.600
MONOMER 1 0.1623 0.2237 0.2585 MONOMER 2 0.8284 0.7661 0.7311
MONOMER 3 0.0093 0.0102 0.0104 0.700 MONOMER 1 0.1449 0.2043 0.2522
MONOMER 2 0.8460 0.7857 0.7374 MONOMER 3 0.0091 0.0100 0.0104 0.800
MONOMER 1 0.1216 0.1770 0.2446 MONOMER 2 0.8698 0.8133 0.7451
MONOMER 3 0.0087 0.0098 0.0103 0.900 MONOMER 1 0.0862 0.1324 0.2349
MONOMER 2 0.9060 0.8585 0.7549 MONOMER 3 0.0078 0.0091 0.0102 0.990
MONOMER 1 0.0195 0.0337 0.2220 MONOMER 2 0.9758 0.9603 0.7679
MONOMER 3 0.0047 0.0060 0.0101
__________________________________________________________________________
TABLE III
__________________________________________________________________________
Ternary Copolymer Kinetics for AA/NaA/MBAM
__________________________________________________________________________
Starting Composition: 22/77/1:AA/NaA/MBAM Molecular MONOMER NAME Q
E Weight
__________________________________________________________________________
MONO 1 AA 1.1500 0.7700 72.05 MONO 2 NAA 0.7100 -0.1200 94.05 MONO
3 MBAM 0.7400 1.0000 154.17
__________________________________________________________________________
OUTPUT UNITS = WEIGHT FRACTIONS MONOMER INSTANTANEOUS CUMULATIVE
CONVERSION MIX POLYMER POLYMER
__________________________________________________________________________
0.000 MONOMER 1 0.2200 0.2836 0.2836 MONOMER 2 0.7700 0.7082 0.7082
MONOMER 3 0.0100 0.0082 0.0082 0.100 MONOMER 1 0.2133 0.2770 0.2804
MONOMER 2 0.7765 0.7147 0.7113 MONOMER 3 0.0102 0.0084 0.0083 0.200
MONOMER 1 0.2058 0.2694 0.2768 MONOMER 2 0.7838 0.7220 0.7148
MONOMER 3 0.0104 0.0086 0.0084 0.300 MONOMER 1 0.1973 0.2607 0.2730
MONOMER 2 0.7921 0.7303 0.7185 MONOMER 3 0.0106 0.0089 0.0085 0.400
MONOMER 1 0.1875 0.2506 0.2687 MONOMER 2 0.8016 0.7401 0.7227
MONOMER 3 0.0109 0.0093 0.0087 0.500 MONOMER 1 0.1761 0.2385 0.2639
MONOMER 2 0.8127 0.7518 0.7273 MONOMER 3 0.0112 0.0097 0.0088 0.600
MONOMER 1 0.1623 0.2235 0.2585 MONOMER 2 0.8262 0.7663 0.7325
MONOMER 3 0.0115 0.0101 0.0090 0.700 MONOMER 1 0.1449 0.2041 0.2522
MONOMER 2 0.8432 0.7852 0.7386 MONOMER 3 0.0119 0.0107 0.0092 0.800
MONOMER 1 0.1217 0.1769 0.2446 MONOMER 2 0.8660 0.8117 0.7460
MONOMER 3 0.0122 0.0115 0.0094 0.900 MONOMER 1 0.0865 0.1325 0.2348
MONOMER 2 0.9009 0.8551 0.7555 MONOMER 3 0.0125 0.0125 0.0097 0.990
MONOMER 1 0.0199 0.0343 0.2220 MONOMER 2 0.9695 0.9537 0.7680
MONOMER 3 0.0106 0.0120 0.0100
__________________________________________________________________________
TABLE IV ______________________________________ Chemical Sequence
Distribution (Diads and Triads) for AA/NaA/MBAM as a function of
conversion. MONOMER A = Acrylic Acid MONOMER B = Sodium Acrylate
MONOMER C = N-N'-Methylene bis Acrylamide
______________________________________ SEQUENCE DISTRIBUTION OUTPUT
AT 0.000 CONVERSION DYAD FRACTIONS AA = 0.07954 BB = 0.38872 CC =
0.00001 AB + BA = 0.52257 AC + CA = 0.00182 BC + CB = 0.00734 TRIAD
FRACTIONS A-CENTERED TOTAL = 0.34174 AAA = 0.01851 BAB = 0.19977
CAC = 0.00000 AAB + BAA = 0.12163 AAC + CAA = 0.00042 BAC + CAB =
0.00139 B-CENTERED TOTAL = 0.65367 ABA = 0.10444 BBB = 0.23116 CBC
= 0.00002 ABB + BBA = 0.31075 ABC + CBA = 0.00294 BBC + CBB =
0.00437 C-CENTERED TOTAL = 0.00459 ACA = 0.00018 BCB = 0.00294 CCC
= 0.00000 ACB + BCA = 0.00146 ACC + CCA = 0.00000 BCC + CCB =
0.00002 ______________________________________ NUMBER DENSITY &
MOL (OR WT) FRACTION DISTRIBUTIONS A B C SEQUENCES SEQUENCES
SEQUENCES LENGTH N.D. W.F. N.D. W.F. N.D. W.F.
______________________________________ 1 0.767 0.167 0.405 0.116
0.998 0.008 2 0.179 0.078 0.241 0.138 0.002 0.000 3 0.042 0.027
0.143 0.123 0.000 0.000 4 0.010 0.008 0.085 0.098 0.000 0.000 5
0.002 0.002 0.051 0.073 0.000 0.000 6 0.001 0.001 0.030 0.052 0.000
0.000 7 0.000 0.000 0.018 0.036 0.000 0.000 8 0.000 0.000 0.011
0.024 0.000 0.000 9 0.000 0.000 0.006 0.016 0.000 0.000 10 0.000
0.000 0.004 0.011 0.000 0.000 15 0.000 0.000 0.000 0.001 0.000
0.000 20 0.000 0.000 0.000 0.000 0.000 0.000
______________________________________ SEQUENCE DISTRIBUTION OUTPUT
AT 0.300 CONVERSION DYAD FRACTIONS AA = 0.07305 BB = 0.40570 CC =
0.00001 AB + BA = 0.51163 AC + CA = 0.00181 BC + CB = 0.00779 TRIAD
FRACTIONS A-CENTERED TOTAL = 0.32962 AAA = 0.01618 BAB = 0.19835
CAC = 0.00000 AAB + BAA = 0.11329 AAC + CAA = 0.00040 BAC + CAB =
0.00141 B-CENTERED TOTAL = 0.66556 ABA = 0.09837 BBB = 0.24741 CBC
= 0.00002 ABB + BBA = 0.31201 ABC + CBA = 0.00300 BBC + CBB =
0.00475 C-CENTERED TOTAL = 0.00481 ACA = 0.00017 BCB = 0.00315 CCC
= 0.00000 ACB + BCA = 0.00147 ACC + CCA = 0.00000 BCC + CCB =
0.00002 ______________________________________ NUMBER DENSITY &
MOL (OR WT) FRACTION DISTRIBUTIONS A B C SEQUENCES SEQUENCES
SEQUENCES LENGTH N.D. W.F. N.D. W.F. N.D. W.F.
______________________________________ 1 0.779 0.165 0.390 0.109
0.998 0.008 2 0.172 0.073 0.238 0.133 0.002 0.000 3 0.038 0.024
0.145 0.122 0.000 0.000 4 0.008 0.007 0.088 0.099 0.000 0.000 5
0.002 0.002 0.054 0.076 0.000 0.000 6 0.000 0.001 0.033 0.055 0.000
0.000 7 0.000 0.000 0.020 0.039 0.000 0.000 8 0.000 0.000 0.012
0.027 0.000 0.000 9 0.000 0.000 0.007 0.019 0.000 0.000 10 0.000
0.000 0.005 0.013 0.000 0.000 15 0.000 0.000 0.000 0.002 0.000
0.000 20 0.000 0.000 0.000 0.000 0.000 0.000
______________________________________ SEQUENCE DISTRIBUTION OUTPUT
AT 0.600 CONVERSION DYAD FRACTIONS AA = 0.06474 BB = 0.42950 CC =
0.00001 AB + BA = 0.49556 AC + CA = 0.00179 BC + CB = 0.00840 TRIAD
FRACTIONS A-CENTERED TOTAL = 0.31308 AAA = 0.01336 BAB = 0.19568
CAC = 0.00000 AAB + BAA = 0.10225 AAC + CAA = 0.00037 BAC + CAB =
0.00141 B-CENTERED TOTAL = 0.68182 ABA = 0.09013 BBB = 0.27083 CBC
= 0.00003 ABB + BBA = 0.31248 ABC + CBA = 0.00306 BBC + CBB =
0.00530 C-CENTERED TOTAL = 0.00511 ACA = 0.00016 BCB = 0.00346 CCC
= 0.00000 ACB + BCA = 0.00147 ACC + CCA = 0.00000 BCC + CCB =
0.00002 ______________________________________ NUMBER DENSITY &
MOL (OR WT) FRACTION DISTRIBUTIONS A B C SEQUENCES SEQUENCES
SEQUENCES LENGTH N.D. W.F. N.D. W.F. N.D. W.F.
______________________________________ 1 0.794 0.163 0.369 0.100
0.998 0.009 2 0.164 0.067 0.233 0.126 0.002 0.000 3 0.034 0.021
0.147 0.119 0.000 0.000 4 0.007 0.006 0.093 0.100 0.000 0.000 5
0.001 0.001 0.058 0.079 0.000 0.000 6 0.000 0.000 0.037 0.060 0.000
0.000 7 0.000 0.000 0.023 0.044 0.000 0.000 8 0.000 0.000 0.015
0.032 0.000 0.000 9 0.000 0.000 0.009 0.022 0.000 0.000 10 0.000
0.000 0.006 0.016 0.000 0.000 15 0.000 0.000 0.001 0.002 0.000
0.000 20 0.000 0.000 0.000 0.000 0.000 0.000
______________________________________ SEQUENCE DISTRIBUTION OUTPUT
AT 0.900 CONVERSION DYAD FRACTIONS AA = 0.05230 BB = 0.47044 CC =
0.00001 AB + BA = 0.46618 AC + CA = 0.00171 BC + CB = 0.00935 TRIAD
FRACTIONS A-CENTERED TOTAL = 0.28546 AAA = 0.00953 BAB = 0.18928
CAC = 0.00000 AAB + BAA = 0.08495 AAC + CAA = 0.00031 BAC + CAB =
0.00139 B-CENTERED TOTAL = 0.70900 ABA = 0.07680 BBB = 0.31285 CBC
= 0.00003 ABB + BBA = 0.31002 ABC + CBA = 0.00308 BBC + CBB =
0.00622 C-CENTERED TOTAL = 0.00553 ACA = 0.00013 BCB = 0.00394 CCC
= 0.00000 ACB + BCA = 0.00144 ACC + CCA = 0.00000 BCC + CCB =
0.00002 ______________________________________ NUMBER DENSITY &
MOL (OR WT) FRACTION DISTRIBUTIONS A B C SEQUENCES SEQUENCES
SEQUENCES LENGTH N.D. W.F. N.D. W.F. N.D. W.F.
______________________________________ 1 0.8l8 0.157 0.335 0.085
0.998 0.010 2 0.149 0.057 0.223 0.113 0.002 0.000 3 0.027 0.016
0.148 0.112 0.000 0.000 4 0.005 0.004 0.099 0.100 0.000 0.000 5
0.001 0.001 0.066 0.083 0.000 0.000 6 0.000 0.000 0.044 0.066 0.000
0.000 7 0.000 0.000 0.029 0.051 0.000 0.000 8 0.000 0.000 0.019
0.039 0.000 0.000 9 0.000 0.000 0.013 0.029 0.000 0.000 10 0.000
0.000 0.009 0.022 0.000 0.000 15 0.000 0.000 0.001 0.004 0.000
0.000 20 0.000 0.000 0.000 0.001 0.000 0.000
______________________________________ SEQUENCE DISTRIBUTION OUTPUT
AT 0.990 CONVERSION DYAD FRACTIONS AA = 0.04565 BB = 0.49584 CC =
0.00001 AB + BA = 0.44716 AC + CA = 0.00162 BC + CB = 0.00972 TRIAD
FRACTIONS A-CENTERED TOTAL = 0.26768 AAA = 0.00765 BAB = 0.18348
CAC = 0.00000 AAB + BAA = 0.07494 AAC + CAA = 0.00027 BAC + CAB =
0.00133 B-CENTERED TOTAL = 0.72669 ABA = 0.06924 BBB = 0.34058 CBC
= 0.00003 ABB + BBA = 0.30715 ABC + CBA = 0.00301 BBC + CBB =
0.00668 C-CENTERED TOTAL = 0.00563 ACA = 0.00011 BCB = 0.00412 CCC
= 0.00000 ACB + BCA = 0 00137 ACC + CCA = 0.00000 BCC + CCB =
0.00002 ______________________________________ NUMBER DENSITY &
MOL (OR WT) FRACTION DISTRIBUTIONS A B C SEQUENCES SEQUENCES
SEQUENCES LENGTH N.D. W.F. N.D. W.F. N.D. W.F.
______________________________________ 1 0.832 0.154 0.313 0.075
0.998 0.010 2 0.140 0.052 0.215 0.103 0.002 0.000 3 0.023 0.013
0.148 0.107 0.000 0.000 4 0.004 0.003 0.101 0.098 0.000 0.000 5
0.001 0.001 0.070 0.084 0.000 0.000 6 0.000 0.000 0.048 0.069 0.000
0.000 7 0.000 0.000 0.033 0.055 0.000 0.000 8 0.000 0.000 0.023
0.043 0.000 0.000 9 0.000 0.000 0.016 0.034 0.000 0.000 10 0.000
0.000 0.011 0.026 0.000 0.000 15 0.000 0.000 0.002 0.006 0.000
0.000 20 0.000 0.000 0.000 0.001 0.000 0.000
______________________________________
TABLE V ______________________________________ BINARY REACTIVITY
RATIOS BINARY REACTION REACTIVITY RATIOS
______________________________________ 1 ACRYLIC ACID R1 = 0.8162 2
SODIUM ACRYLATE R2 = 0.5549 1 ACRYLIC ACID R1' = 1.8552 2 MBAM R2'
= 0.5113 1 SODIUM ACRYLATE R1" = 0.8388 2 MBAM R2" = 0.3401
______________________________________
Additional advantages and modifications will readily occur to those
skilled in the art. The invention in its broader aspects is,
therefore, not limited to the specific details, representative
apparatus and illustrative examples shown and described.
Accordingly, departures may be made from such details without
departing from the spirit and scope of the general inventive
concept as defined by the appended claims and their
equivalents.
* * * * *